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Physical Review

Eigenvalues of Fermion Density Matrices

Fukashi Sasaki

Published Jun 7, 1965• 143 citations
Publisher

Abstract

The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be $n$ for a system of $2n$ or $2n+1$ identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.

DOI
10.1103/physrev.138.b1338

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Physical Review•Jul 29, 1966•67 citations
The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be nnn for a system of 2n2n2n or 2n+12n+12n+1 identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.
For a system of NNN identical particles in a pure bound state, the density matrices of positive orders ppp and $N\ensuremath{-}p$ have the same nonzero eigenvalues with the same multiplicities. If the number of nonzero eigenvalues is finite, these density matrices are unitarily equivalent.
Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``PPP'' and ``QQQ'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for NNN-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''
A division of the 2-matrix Γ, for a system of N identical fermions, into 2 parts is proposed in such a way that Γ has the same 1-matrix γ as does Γ and Γ has a vanishing 1-matrix (and therefore vanishing trace). This is accomplished by evaluating the natural spin orbital (NSO) occupation numbers Γ of Γ from (where the ni are NSO occupation numbers of γ); and letting γ0 have the same NSO's as does γ. The physical interpretation of Γ should be easier with this decomposition, since Γ is completely determined by γ and has a Hartree-Fock-like form, while Γ corrects the pair density of Γ without disturbing its 1-density.
For a quantum-mechanical system of NNN identical fermions, the NNN-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric NNN-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, NNN, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be NNN-representable is that its natural spin-orbital occupation numbers be equal in pairs.